CHAPTER 2.
20
Comparing 2.7, 2.9, and 2.10 the following two equation can be written: ZZ n(r)n(r0 ) δEH [n(r)] = drdr0 , vH (r) = 0 δn(r) |r − r |
(2.11)
δEXC [n(r)] . (2.12) δn(r) After some more calculation and writing kinetic energy separately an equation very similar vXC (r) =
to Fock equation appears. This is the Kohn-Sham equation. 1 [− ∇2 + vKS (r)]Ďˆi (r) = i Ďˆi (r), 2
(2.13)
where i is the Lagrange multipliers corresponding to the orthonormality of the N singleparticle states Ďˆi (r). The density of electron is n(r) =
N X
|Ďˆi (r)|2 .
(2.14)
i=1
The non-interacting kinetic energy TS [n(r)] looks N Z 1X TS [n(r)] = − Ďˆi∗ (r)∇2 Ďˆi (r)d(r). 2 i=1
(2.15)
Thus in Kohn-Sham theory, in order to handle the kinetic energy in an exact manner, N equations have to be solved to obtain the set of Lagrange multipliers { i } whereas in Thomas Fermi model only one equation have to be solved to determine the multiplier Âľ. The main aim of Kohn-Sham theory was to make the unknown contribution to the total energy of the non-interacting system as small as possible. Though the contribution of exchange-correlation energy is very small, still it is an important contribution as the binding energy of many systems is about the same size as EXC [n(r)]. So an accurate description of exchange-correlation energy is very much crucial in DFT.
2.1.4
Exchange-Correlation: With Dirac Exchange
As described in the previous section, the accuracy of the exchange-correlation energy is the central importance in DFT. The history of EXC stem from the addition of exchange